Circular Motion Angular Variables Arc Length The arc

Circular Motion: Angular Variables • Arc Length: The arc length s is related to the angle q (in radians = rad) as follows: (360 o = 2 p rad) • Angular Displacement and Angular Velocity: (radians/second) • Tangential Velocity and Angular Velocity: Tangential Velocity • Period and Frequency (constant w): (revolutions/second) R. Field 2/5/2013 University of Florida PHY 2053 1

Circular Motion: Radial Acceleration • Radial (centripetal) Acceleration: Centripetal acceleration = “toward the center” Radial Acceleration Magnitude = (vt)2/r Direction = toward the center of the circle R. Field 2/5/2013 University of Florida PHY 2053 2

Example Problem • A puck of mass m slides in a circle of radius r = 0. 5 m on a frictionless table while attached to a hanging cylinder of mass M = 2 m by a cord through a hole in the table. What speed of the mass m keeps the cylinder at rest? Solution: Answer: FT FT Mg R. Field 2/5/2013 University of Florida PHY 2053 3

Example: Conical Pendulum • A stone of mass m is connected to a cord with length L and negligible mass. The stone is undergoing uniform circular motion in the horizontal plane. If the cord makes an angle f with the vertical direction, what is the period of the circular motion? x-component: y-component: R. Field 2/5/2013 University of Florida PHY 2053 4

Exam 1 Spring 2012: Problem 20 • A conical pendulum is constructed from a stone of mass M connected to a cord with length L and negligible mass. The stone is undergoing uniform circular motion in the horizontal plane as shown in the figure. If the cord makes an angle q = 30 o with the vertical direction and the period of the circular motion is 4 s, what is the length L of the cord (in meters)? Answer: 4. 59 % Right: 34% R. Field 2/5/2013 University of Florida PHY 2053 5

Example Problem: Unbanked Curves • A car of mass M is traveling in a circle with radius R on a flat highway with speed v. If the static coefficient of friction between the tires and the road is ms, what is the maximum speed of the car such that it will not slide? x-component: y-component: R. Field 2/5/2013 University of Florida PHY 2053 6

Exam 2 Spring 2011: Problem 4 • Near the surface of the Earth, a car is traveling at a constant speed v around a flat circular race track with a radius of 50 m. If the coefficients of kinetic and static friction between the car’s tires and the road are mk = 0. 1, ms = 0. 4, respectively, what is the maximum speed the car can travel without slipping? Answer: 14 m/s % Right: 74% R. Field 2/5/2013 University of Florida PHY 2053 7

Example Problem: Banked Curves • If the car in the previous problem is traveling on a banked road (angle q), what is the maximum speed of the car such that it will not slide? x-component: y-component: R. Field 2/5/2013 University of Florida PHY 2053 8

Rolling Without Slipping: Rotation & Translation • If a cylinder of radius R rolls without slipping along the x-axis then: Translational Speed R. Field 2/5/2013 University of Florida Rotational Speed PHY 2053 9